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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.7.11
Chapter 10, Problem 10.7.11

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
∑ (from k = 1 to ∞) ((-1)ᵏ⁺¹) × ((10k³ + k) / (9k³ + k + 1))ᵏ

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Identify the general term of the series: \(a_k = (-1)^{k+1} \left( \frac{10k^3 + k}{9k^3 + k + 1} \right)^k\).
Since the series has terms raised to the power \(k\), the Root Test is a natural choice. Recall the Root Test uses the limit \(L = \lim_{k \to \infty} \sqrt[k]{|a_k|}\).
Calculate \(\sqrt[k]{|a_k|}\). Because of the absolute value, the \((-1)^{k+1}\) disappears, so \(\sqrt[k]{|a_k|} = \sqrt[k]{\left( \frac{10k^3 + k}{9k^3 + k + 1} \right)^k} = \frac{10k^3 + k}{9k^3 + k + 1}\).
Evaluate the limit \(L = \lim_{k \to \infty} \frac{10k^3 + k}{9k^3 + k + 1}\). To do this, divide numerator and denominator by \(k^3\) to simplify the expression.
Interpret the result of the limit \(L\): if \(L < 1\), the series converges absolutely; if \(L > 1\), the series diverges; if \(L = 1\), the Root Test is inconclusive.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ratio Test

The Ratio Test determines the convergence of a series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
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Ratio Test

Root Test

The Root Test analyzes the nth root of the absolute value of the nth term of a series. If the limit of this root is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive. It is especially useful for terms raised to the nth power.
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Root Test

Absolute Convergence

A series converges absolutely if the series of absolute values of its terms converges. Absolute convergence implies convergence regardless of term signs, which is important when applying tests like the Ratio or Root Test to series with alternating signs.
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Choosing a Convergence Test
Related Practice
Textbook Question

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.


{(−1)ⁿ / 2ⁿ}

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Textbook Question

What test is advisable if a series involves a factorial term?

Textbook Question

51–56. {Use of Tech} Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.


aₙ₊₁ = 4aₙ + 1 a₀ = 1

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Textbook Question

84–87. {Use of Tech} Sequences by recurrence relations

The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.


a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.

b.Use analytical methods to find the limit of the sequence.


aₙ₊₁ = 2aₙ(1 − aₙ);a₀ = 0.3

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Textbook Question

{Use of Tech} For what value of r does

∑ (k = 3 to ∞) r²ᵏ = 10?

Textbook Question

What comparison series would you use with the Comparison Test to determine whether

∑ (k = 1 to ∞) 1 / (k² + 1) converges?

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